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Trump in childish Twitter war with Kim

US President Donald Trump is exchanging school yard taunts with North Korea's Kim Jong Un.

In a response to North Korea calling Trump's speech in South Korea "reckless remarks by an old lunatic," Trump tweeted from Hanoi on Sunday morning: "Why would Kim Jong-un insult me by calling me 'old,' when I would NEVER call him 'short and fat?"'

Trump goes on to say sarcastically, "Oh well, I try so hard to be his friend - and maybe someday that will happen!"

Trump has been working to rally global pressure against North Korea's nuclear weapons program on a trip to Asia.

That includes a stern speech delivered in South Korea's National Assembly on Tuesday, in which he said: "Do not underestimate us. And do not try us. ... The weapons you're acquiring are not making you safer, they are putting your regime in grave danger. Every step you take down this dark path increases the peril you face."

On Saturday, Kim's government accused Trump of trying to demonise North Korea.

"Reckless remarks by an old lunatic like Trump will never scare us or stop our advance," the North's Foreign Ministry said in a statement.

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"On the contrary ... it pushes us to speed up the effort to complete our nuclear force."

Asked at news conference with Vietnam's president on Sunday if he could become friends with the North Korean leader, Trump replied it "might be a strange thing to happen but it's certainly a possibility".

He said he doesn't know if that friendship will develop, but added it would be "very, very nice if it did".